Building a class of fuzzy equivalence relations

In this paper, we propose a practical method, given a strict triangular norm with a convex additive generator, for deriving a fuzzy equivalence relation whose reflexivity condition generalizes Ruspini's definition of fuzzy partitions. The properties of the relations, their comparison, their transitivity, the construction of fuzzy equivalence relations on cartesian products are presented. A large part of the paper is devoted to applications with fuzzy partitions defined on the real line. Several examples, including the fairy-tale problem from De Cock and Kerre [On (un)suitable fuzzy relations to model approximate equality, Fuzzy Sets and Systems 133 (2) (2003) 137-153], the comparison of colored objects and comfort situations are proposed.

[1]  Didier Dubois,et al.  Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities , 2004, Reliab. Comput..

[2]  Bernard De Baets,et al.  T -partitions , 1998 .

[3]  W. Pedrycz.  Why triangular membership functions , 1994 .

[4]  Frank Klawonn,et al.  Should fuzzy equality and similarity satisfy transitivity? Comments on the paper by M. De Cock and E. Kerre , 2003, Fuzzy Sets Syst..

[5]  Witold Pedrycz,et al.  Fuzzy equalization in the construction of fuzzy sets , 2001, Fuzzy Sets Syst..

[6]  Radko Mesiar,et al.  Fuzzy partitions and their entropy , 1996 .

[7]  Ulrich Bodenhofer,et al.  A note on approximate equality versus the Poincaré paradox , 2003, Fuzzy Sets Syst..

[8]  Vladimír Janis,et al.  Resemblance is a nearness , 2003, Fuzzy Sets Syst..

[9]  Laurent Foulloy,et al.  Fuzzy sensor aggregation: application to comfort measurement , 1994 .

[10]  T. Sudkamp.  On probability-possibility transformations , 1992 .

[11]  Enrique H. Ruspini,et al.  A New Approach to Clustering , 1969, Inf. Control..

[12]  H. J. Skala,et al.  Aspects of vagueness , 1984 .

[13]  Martine De Cock,et al.  On (un)suitable fuzzy relations to model approximate equality , 2003, Fuzzy Sets Syst..

[14]  L. Valverde.  On the structure of F-indistinguishability operators , 1985 .

[15]  Bernard De Baets,et al.  The functional equations of Frank and Alsina for uninorms and nullnorms , 2001, Fuzzy Sets Syst..

[16]  Laurent Foulloy,et al.  A fuzzy colour sensor , 1994 .

[17]  Adolfo R. de Soto,et al.  Modelling a linguistic variable as a hierarchical family of partitions induced by an indistinguishability operator , 2001, Fuzzy Sets Syst..

[18]  D. Dubois,et al.  FUZZY NUMBERS: AN OVERVIEW , 1993 .

[19]  J. Recasens,et al.  Fuzzy Equivalence Relations: Advanced Material , 2000 .

[20]  J. Bezdek,et al.  Fuzzy partitions and relations; an axiomatic basis for clustering , 1978 .

[21]  L. Finkelstein,et al.  REPRESENTATION BY SYMBOL SYSTEMS AS AN EXTENSION OF THE CONCEPT OF MEASUREMENT , 1975 .

[22]  S. Ovchinnikov.  An Introduction to Fuzzy Relations , 2000 .

[23]  Frank Klawonn,et al.  Equality Relations as a Basis for Fuzzy Control , 1993 .

[24]  Lotfi A. Zadeh,et al.  Quantitative fuzzy semantics , 1971, Inf. Sci..

[25]  Martine De Cock,et al.  Why fuzzy -equivalence relations do not resolve the Poincaré paradox, and related issues , 2003, Fuzzy Sets Syst..

[26]  Dionís Boixader,et al.  On the relationship between T-transitivity and approximate equality , 2003, Fuzzy Sets Syst..

[27]  B. Baets,et al.  Metrics and T-Equalities , 2002 .